Quiver moduli turns representations of a directed graph into algebraic geometry. Fix a quiver and a dimension vector. All representations with those dimensions form an affine variety. A product of general linear groups acts by change of bases at the vertices. King stability then selects the representations that should survive, and geometric invariant theory forms their quotient.
This is one of the cleanest places where several BTT Representation Mathematics threads meet: quiver representations supply the objects, GIT supplies the quotient, and Hall algebras show how extension theory turns the same category into quantum multiplication.
Representation varieties · Base-change action · King stability · Worked examples · GIT quotient · Fine and coarse moduli · Practice
Fix a quiver and dimension vector
Let Q=(Q_0,Q_1) be a finite quiver with vertices Q_0 and arrows Q_1. A dimension vector is a tuple d=(d_i)_{i∈Q_0} of nonnegative integers.
A representation M of dimension d assigns a vector space M_i≈C^{d_i} to each vertex and a linear map M_a:M_{s(a)}→M_{t(a)} to each arrow.
Once bases are fixed, every arrow map becomes a matrix. Thus all such representations live in one affine space.
The representation variety
Define
Rep(Q,d)=⊕_{a:i→j} Hom(C^{d_i},C^{d_j}).
Its dimension is
dim Rep(Q,d)=Σ_{a:i→j} d_i d_j.
For the one-arrow quiver 1→2 with dimension vector (m,n), the representation variety is simply the matrix space Mat_{n×m}(C), of dimension mn.
Change of basis is the group action
The base-change group is
G_d=∏_{i∈Q_0} GL_{d_i}(C).
It acts on a representation by
(g_i)_i·(M_a)_a=(g_{t(a)}M_ag_{s(a)}^{-1})_a.
Two points of Rep(Q,d) lie in the same G_d-orbit exactly when the corresponding quiver representations are isomorphic.
Therefore the classification problem is an orbit problem.
Why the raw orbit space is inadequate
Orbits can fail to be closed. A family of representations may degenerate to a more reducible representation in the orbit closure.
The affine quotient Rep(Q,d)//G_d identifies representations whose orbit closures meet. In many acyclic examples the ring of ordinary invariants is too small to distinguish useful objects.
King’s insight is to introduce a character of G_d and use projective GIT stability.
King stability
Choose an integer weight θ=(θ_i)_{i∈Q_0} satisfying
θ·d=Σ_i θ_i d_i=0.
For a representation M with dimension vector d, define θ(M)=θ·dim M.
Using one common convention, M is θ-semistable if
θ(N)≥0
for every nonzero proper subrepresentation N⊂M, and θ-stable if the inequalities are strict.
Some references use the opposite sign. What matters is consistency between the character χ_θ and the inequality.
The character defining the linearization
The weight θ defines a character
χ_θ((g_i))=∏_i det(g_i)^{θ_i}
up to the sign convention chosen for stability.
Because θ·d=0, scalar matrices acting simultaneously at all vertices lie in the kernel of this character. This compatibility is necessary because the diagonal scalar subgroup acts trivially on the representation variety itself.
Worked example: one arrow 1→2 with d=(1,1)
A representation is a scalar a:C→C.
Choose θ=(−1,1), so θ·d=0. If a≠0, there is no subrepresentation of dimension (1,0), because the nonzero map sends the first vertex into the second. There is a subrepresentation of dimension (0,1), whose θ-value is +1.
Thus nonzero maps are θ-stable in this convention.
If a=0, the representation has a subrepresentation of dimension (1,0), with θ-value −1, so it is unstable.
The stable locus is therefore C×.
The quotient in the one-arrow example
G_d=GL_1×GL_1 acts by
(g_1,g_2)·a=g_2ag_1^{-1}.
Every nonzero scalar can be transformed to 1. Therefore all stable representations are isomorphic.
The moduli space consists of one point.
This is a useful example where the stable locus is one-dimensional but the quotient is zero-dimensional because the group action removes the coordinate freedom.
A rank interpretation
For one arrow C^m→C^n, the base-change orbits are classified by matrix rank.
Appropriate stability parameters can force injectivity or surjectivity conditions, depending on dimensions and sign convention.
After quotienting, Grassmannians naturally appear: for instance, injective maps C^m→C^n modulo GL_m identify m-dimensional subspaces of C^n.
Thus ordinary Grassmannians arise as quiver moduli in elementary cases.
Slope-style stability
In many quiver problems one packages θ with a positive rank function and defines a slope μ(M). Stability then resembles vector-bundle stability: every proper subobject must have smaller or larger slope according to convention.
This makes Harder–Narasimhan filtrations available. Every representation can be decomposed into successive semistable pieces of ordered slopes under suitable hypotheses.
Stability therefore organizes not only which points survive in moduli but also how unstable objects decompose canonically.
Jordan–Hölder and S-equivalence
A semistable representation can have a Jordan–Hölder filtration whose stable factors all have the same slope.
The associated graded object is polystable. Two semistable objects are S-equivalent when their associated graded polystable representations are isomorphic.
The GIT quotient classifies S-equivalence classes of semistable representations rather than literal isomorphism classes of every semistable point.
Stable points, by contrast, typically correspond to genuine isomorphism classes.
The moduli quotient
King’s construction forms
M_θ(Q,d)=Rep(Q,d)^{ss}_θ // G_d
as a projective GIT quotient relative to the chosen character in the standard finite-quiver setup.
Its closed points correspond to S-equivalence classes of θ-semistable representations.
The stable locus produces an open subspace parametrizing stable isomorphism classes with better geometric behavior.
Expected dimension
At a stable representation with scalar automorphisms only, the expected local dimension is often
dim Rep(Q,d)−dim G_d+1.
The +1 accounts for the diagonal scalar subgroup acting trivially.
For an acyclic quiver, this equals
1−⟨d,d⟩
using the Euler form ⟨d,d⟩=Σ_i d_i²−Σ_{a:i→j}d_id_j.
Schur roots
A dimension vector d is a Schur root if there exists a representation M of dimension d with End(M)=C.
Stable representations are Schur objects because any endomorphism has eigenspace kernels/images that would otherwise violate stability; over an algebraically closed field, the endomorphism ring reduces to scalars.
Thus nonempty stable moduli imposes strong representation-theoretic conditions on d.
Wall and chamber structure
The condition θ·d=0 restricts θ to a hyperplane. Potential destabilizing dimension vectors e define walls θ·e=0.
Inside a chamber avoiding all relevant walls, the stable/semistable loci remain unchanged.
Crossing a wall changes which subrepresentations are permitted and can produce birational transformations of moduli spaces.
This is the quiver version of variation of GIT.
Fine versus coarse moduli
A fine moduli space carries a universal family whose pullback classifies every family in the moduli problem.
A coarse moduli space only corepresents the classification of isomorphism or S-equivalence classes and need not carry a universal family.
Stable quiver moduli can often admit universal families after imposing indivisibility or choosing suitable rigidification. In general, scalar automorphisms can obstruct a universal family and lead naturally to moduli stacks.
Therefore “moduli space” should not automatically be assumed fine.
Moduli stacks retain automorphisms
The quotient stack [Rep(Q,d)/G_d] remembers stabilizer groups. The coarse GIT quotient forgets most automorphism data.
Stacks are therefore the more faithful quotient object when families and automorphisms matter.
This distinction mirrors categorification more broadly: forgetting automorphisms can simplify the space while losing structure.
From quiver moduli to Nakajima varieties
Nakajima quiver varieties enrich the quiver by doubling arrows, adding framing spaces and imposing moment-map equations before taking a stability quotient.
The resulting spaces carry symplectic geometry and realize representations of Kac–Moody algebras on their homology or K-theory.
Thus ordinary King moduli is the natural GIT doorway into the final article of Batch 11.
A verification workflow
- 1. Fix Q and dimension vector d.
- 2. Write Rep(Q,d) explicitly.
- 3. Write the base-change group and its action.
- 4. Choose θ with θ·d=0.
- 5. State the sign convention for stability.
- 6. Enumerate possible subrepresentation dimension vectors.
- 7. Test θ on each destabilizing subobject.
- 8. Distinguish stable, semistable and polystable.
- 9. Interpret the quotient as S-equivalence classes.
- 10. Check whether the desired moduli problem is fine, coarse or stacky.
Common mistakes
- Forgetting θ·d=0.
- Testing arbitrary subspaces instead of subrepresentations.
- Mixing stability sign conventions.
- Equating semistable isomorphism classes with quotient points.
- Ignoring scalar stabilizers in dimension counts.
- Calling every moduli space fine.
- Ignoring walls in θ-space.
- Confusing representation variety dimension with quotient dimension.
Practice questions
1. Define Rep(Q,d). 2. What is G_d? 3. Why do its orbits classify isomorphism classes? 4. What equation must θ satisfy?
5. For 1→2 with d=(1,1) and θ=(−1,1), which maps are stable? 6. What is the quotient of the stable locus? 7. What does S-equivalence identify? 8. State the expected stable-moduli dimension formula.
9. What is a Schur root? 10. What defines a stability wall? 11. Why can a universal family fail to exist? 12. What extra structure do quotient stacks retain?
Worked answers
1. The direct sum of Hom spaces for all arrows with fixed vertex dimensions.
2. ∏_i GL_{d_i}, acting by changes of bases at the vertices.
3. An isomorphism is exactly a compatible family of invertible basis changes.
4. θ·d=0.
5. The nonzero maps.
6. One point, because every nonzero scalar map is isomorphic to the identity map.
7. Semistable objects with the same polystable associated graded.
8. dim Rep(Q,d)−dim G_d+1, under the stated stable scalar-automorphism assumptions.
9. A dimension vector admitting a representation with scalar endomorphism algebra.
10. An equation θ·e=0 for a potentially destabilizing subdimension vector e.
11. Nontrivial automorphisms, especially scalars, can obstruct descent of a universal representation.
12. Stabilizer/automorphism groups and family-level quotient information.
Sources and further study
A. D. King’s paper Moduli of Representations of Finite-Dimensional Algebras established the GIT stability criterion for quiver representations. Reineke’s surveys give accessible treatments of quiver moduli, while Crawley-Boevey and others connect stability with roots, moment maps and quiver varieties.
Representation Mathematics — Batch 11
Begin with Geometric Invariant Theory. Move from abelian quiver categories into Derived Categories and Exceptional Collections. Then impose moment maps and framing in Nakajima Quiver Varieties. Return to the BTT Mathematics Learning Hub.